Decompositions of 2 × 2 matrices over local rings

An element a of a ring R is called perfectly clean if there exists an idempotent e ∈ comm(a) such that a − e ∈ U(R). A ring R is perfectly clean in case every element in R is perfectly clean. In this paper, we completely determine when every 2 × 2 matrix and triangular matrix over local rings are perfectly clean. These give more explicit characterizations of strongly clean matrices over local rings. We also obtain several criteria for a triangular matrix to be perfectly J-clean. For instance, it is proved that for a commutative local ring R, every triangular matrix is perfectly J-clean in Tn(R) if and only if R is strongly J-clean.